% 标题,中文
\ctitle{关于微分方程的辛方法和李群方法研究}

% 作者,中文
\cauthor{卢燚}

% 学科,中文,本科生不需要
\csubject{数学}

% 导师姓名,中文
\csupervisor{蒋耀林~~教授}

% 关键词,中文.用全角分号「；」分割
% 研究生的应首先从《汉语主题词表》中摘选
\ckeywords{波形松弛；辛方法；辛波形松弛方法；李群方法；微分方程}

% 提交日期,本科生不需要
%\cproddate{\the\year 年\the\month 月}
\cproddate{2016 年 12 月}

% 论文类型,中文,本科生不需要
% 从理论研究、应用基础、应用研究、研究报告、软件开发、设计报告、案例分析、调研报告、其它中选择
\ctype{应用基础}

% 论文标题,英文
\etitle{Research on Symplectic Method and Lie Group Method of Differential Equations}

% 作者姓名,英文
\eauthor{Yi Lu}

% 学科,英文,本科生不需要
\esubject{Mathematics}

% 导师姓名,英文
\esupervisor{Prof. Yao-Lin Jiang}

% 关键词,英文.用半角分号和一个半角空格「; 」分割
\ekeywords{Waveform relaxation; Symplectic method; Symplectic waveform relaxation method; Lie group method; Differential equations}

% 学科门类,英文
% 从Philosophy（哲学）、Economics（经济学）、Law（法学）、Education（教育学）、Arts（文学）、
%   Science（理学）、Engineering Science（工学）、Medicine（医学）、Management Science（管理学）中选择
\ecate{Science}

% 提交日期,英文,本科生不需要
% 应当和 cproddate 保持一致
%\eproddate{\monthname{\month}\ \the\year}
\eproddate{December\ 2016}

% 论文类型,英文,本科生不需要
% 从Theoretical Research（理论研究）、Application Fundamentals（应用基础）、Applied Research（应用研究）、
%   Research Report（研究报告）、Software Development（软件开发）、Design Report（设计报告）、
%   Case Study（案例分析）、Investigation Report（调研报告）、其它（Other）中选择
\etype{Application Fundamentals}

% 摘要,中文.段间空行
\cabstract{

高性能计算机以及集群的迅猛发展,使数值模拟成为了科学实验和工程实验领域里一个非常重要的工具,
一方面加速了产品产出的速度,另一方面给科学研究提供了极大便利.
随之而来的是不断涌现的新型数值计算方法和对更高精度更高效数值算法的需求.这些年来,数值计算领域的研究如雨后春笋般发展,日新月异.
数值计算的应用,尤其是微分方程数值模拟的应用更是遍及各个重要的工程领域乃至经济金融领域.随着工程和军
事上应用的蓬勃发展,对数值计算的精准性和时效性的要求日益迫切.在大规模集成电路模拟领域以及航空航天
领域,有一类相当重要的问题需要快速准确地求解,这使得针对快速高效数值计算方法的研究变得非常重要.

本文主要针对几类微分方程,研究讨论了辛(Symplectic)方法、辛波形松弛方法、李群方法和李点对称方法,
提出了几种便于实现并且计算速度快的计算方法.详细的研究内容和主要结论如下:

(一) 对于一类特殊类型的电报方程,应用辛方法进行计算.辛方法适用于哈密尔顿系统等具有特殊结构的
系统,然而电报方程本身不具有这样的结构.为了用辛方法计算,以获得更好的数值结果,我们首先使用了一个变换对其进行修正,然后再使用辛方法进行
求解,最后再用逆变换将数值结果变成原方程的数值结果.该算法的设计采用了和傅立叶变换方法类似的思想,即先变换再
求解再逆变换的思想.整体上讲,该方法利用了辛方法的优势,较好地计算了较长时间的电报方程的数值解.

(二) 对拓展 QZK 方程,利用李点对称方法,进行了一些方程的性质分析,并给出了方程的一些约化.李点对称
方法的思想是将李群作用到微分方程上,通过构造特定的变换,进而得到一些方程的性质甚至是精确解.我们根据 Ibragimov
新守恒律定理构造了拓展 QZK 方程的守恒律.同时,找到了一个最优系统的一维子代数,然后对最优子代数
系统进行了相似约化,将 $(2+1)$ 维拓展 QZK 方程约化为含有两个独立变量的线性偏微分方程.

(三) 基于哈密尔顿系统的辛方法,结合波形松弛方法,首次提出了辛波形松弛方法的概念,降低了原始问题隐式辛方法求解过程中的计算复杂程度,同时较辛方法相比缩短了计算时间.该方法利用了波形松弛方法解耦计算的优势,以及辛方法较好的长时间计算稳定性,将二种方法有机结合,达到一举两得的效果.该算法里波形松弛方法用来简化辛方法的求解
过程,辛方法用来指导波形松弛方法中分裂函数的选取,二者相得益彰.在此基础上,对于流形上的李群方程,利用
辛波形松弛方法的设计思想,提出了隐式 RK-MK 方法的波形松弛改造格式.李群方程属于流形上的微分方程,
原本在流形上的方程的解随着时间演变依然在流形上.但是使用传统的数值解法,数值解往往会脱离流形, RK-MK 方法就是使得数值结果依然落到流形上的一类方法,
该方法利用指数映射做到了这一点.然而,对于一些流形上保结构的问题,对 RK-MK 的
隐式格式需求依然会带来求解上的困难,故我们利用辛波形松弛方法的思想,设计出便于计算的格式,较好地解决了此问题.

在本文中,我们提出了一类电报方程的辛方法,哈密尔顿系统的辛波形松弛方法,以及该方法在流形上的一个
改造,并用李群方法来约化偏微分方程.在数值算例上显式出的较好的效果,约化的方程形式得到的化简,都说明了该
算法的有效性.最后,对工作进行了总结并对进一步研究进行了展望.

}

% 摘要,英文.段间空行
\eabstract{

At present, the rapid advancement of high performance computers as well as workstations has pushed numerical
simulation to a highly demanded position both in numerical analysis for science research and in numerical simulation
for industrial production, which reduces the cycle of science research and production. What it brings is numerous
approaches of numerical methods, the refreshing demands of high precision and efficient numerical algorithms as well as
numerous studies in this area. Scientific computation, especially computation for differential equations has been widely applied
in many areas and even in economical areas. As the applications in industrial and
military areas increased, the demands for high precision and real time numerical methods are urgent. Tremendous problems need to be solved
in large scale integrated circuits and space science fields, which emphasizes the importance of fast and efficient computing methods.

In this dissertation, our study focuses on symplectic method, symplectic waveform relaxation method, Lie group
method and Lie point symmetry method. Contributions and major conclusions are listed as follows:

(1) The symplectic method is applied to certain kind of telegraph equations. The symplectic method is often used to solve
Hamiltonian systems, which is not what normal telegraph equations can fit. Here, we take a pre-transform and then
solve it with the symplectic method, and afterwards drag the numerical solution back to the original telegraph equation
by an inverse transform. This idea resembles the idea of the Fourier transform method. They all solve equations in the
frequency domain and apply an inverse transform to bring the solution to the time domain. This method takes
advantage of the symplectic method, so it is expected to have long-term numerical results.

(2) For extended quantum Zakharov-Kuznetsov equations, we get some properties of this equation by the Lie point
symmetry method, and get a reduced equation. The Lie point symmetry method is a method that applies Lie group to differential
equations. With certain transforms, we can get some information of the equation as well as the solution. By the
theory of Ibragimov, a new conservation law is constructed for extended QZK equations. We find an optimal system of one dimensional
Lie subalgebras and reduce the equation with it so that the $(2+1)$ dimensional extended QZK equation is reduced
to a partial differential equation with two independent variables.

(3) Combining the symplectic method and the waveform relaxation method~(WR), we first propose a new method called
symplectic waveform relaxation for Hamiltonian systems, which makes it possible to compute a Hamiltonian system
``with waveform relaxation method''. The advantage is then a faster and easier method for Hamiltonian systems. It takes
advantage of both decoupling feature of the waveform relaxation method and long-term computation
feature of the symplectic method, which is also an organic combination for both. The waveform relaxation method is used to
simplify the symplectic method while the symplectic method is used to instruct how to choose splitting function of the waveform
relaxation method. After that, for Lie group equations on manifolds, based on
the idea of the symplectic waveform relaxation method, we come up with a new method of modifying the RK-MK method
for Lie group equations. Lie group equations are equations that depict flows on manifolds, whose solution lies on manifold.
Classical numerical methods cannot guarantee that the solution lies on the manifold, which is far from expectation.
The RK-MK method is then such a remedy, which uses exponential mapping to drag the solution back to
the manifold. However, some RK-MK schemes cannot ensure another requirement of structure preserving,
which leads to the need of implicit RK-MK. We propose a method to overcome the difficulty of solving with an implicit RK-MK method.

In conclusion, in this dissertation, we use the symplectic method to solve telegraph equations, we use the Lie point symmetry method
to reduce the extended QZK equation and propose a simple method for implicit numerical schemes by waveform relaxation for
Hamiltonian systems as well as for Lie group equations. Numerical experiments show the efficiency of
our numerical methods. The results also show that the reduced equation is easier than the original problems. At last, we give a conclusion
and an outlook.

}
